
for all \( \varphi \in C_0^{\infty }(\mathbb {R}) \otimes \mathcal {D}_{\circ }^b \) ,
where \( \mathcal {D}_{\circ }^b \)  is the set consisting of bounded, local, and smooth functions on \( \mathfrak {S}\)  [1]}. We write \( \mathfrak {s}= \sum _i \delta _{s^i}\) .
It has been proved [2]} that \( \mu \)  has a logarithmic derivative such that,
strongly in \( L_{\mathrm {loc}}^{2} (\mathbb {R}\times \mathfrak {S}, \mu ^{[1]})\) ,
\(\mathfrak {d}^{\mu }(s , \mathfrak {s}) & = 2 \lim _{R\rightarrow \infty }\sum _{{s^i}\in S_{R}} \frac{1}{s -{s^i}}= 2 \lim _{R\rightarrow \infty }\sum _{ | s -{s^i}| < R} \frac{1}{s -{s^i}}.\) 
